We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$, and prove the following Quasi-Additivity Law: If the total extremal width $\sum \mathcal{W}(S\smallsetminus D_i)$ is big enough (depending on $N$) then it is comparable with the extremal width $\mathcal{W} (S,\cup D_i)$ (under a certain ``separation assumption'') . We also consider a branched covering $f: U\rightarrow V$ of degree $N$ between two disks that restricts to a map $\Lambda\rightarrow B$ of degree $d$ on some disk $\Lambda \Subset U$. We derive from the Quasi-Additivity Law that if $\mod(U\smallsetminus \Lambda)$ is sufficiently small, then (under a ``collar assumption'') the modulus is quasi-invariant under $f$, namely $\mod(V\smallsetminus B)$ is comparable with $d^2 \mod(U\smallsetminus \Lambda)$. This Covering Lemma has important consequences in holomorphic dynamics which will be addressed in the forthcoming notes.